(Solved): 4. Gravity and Altitude (5 points): Acceleration due to the gravity of the Earth is given by a=r2G ...

4. Gravity and Altitude (5 points): Acceleration due to the gravity of the Earth is given by $$a=\frac{GM}{r^2}$$ where $$M?5.97?10^{24}\mathrm{kg}$$ is the mass of the Earth and Newton's gravitational constant is $$G?6.674?10^{?11}{\mathrm{Nm}}^2/{\mathrm{kg}}^2$$. For objects near the surface of the Earth the acceleration due to gravity is nearly constant. Consider an object at an altitude $$h$$ above the International space station orbiting Earth's surface (i.e. $$r=R+h$$ ) where $$R?6371\mathrm{km}$$ is the radius of at an altitude of $$?400\mathrm{km}$$ above the Earth and $$h?R$$. the surface of the Earth. (a) (3 points) Show that $$a?a_0\left(1?2\left(\frac{h}{R}\right)+3{\left(\frac{h}{R}\right)}^2+?\right)$$ where $$a_0?GM/R^2$$. What is $$a_0$$ in SI units? (b) (2 points) How large are the first and second corrections at the altitude of a long-distance commerical flight? (take $$h?10\mathrm{km}$$ ). How what about on the International Space Station (ISS) at $$h?400\mathrm{km}$$ ?